Since you did not work tyhe required length of time for the company and you had notice that you must do so in order to receive an experience letter, you are not entitled to such a letter. Perhaps if you speak to your employer and explain why you were onjly able to give 1 week's notice, the former employer will agree to furnish an exnperience letter.
Assuming that you are referring to some written confirmation that you worked at the company, as opposed to the "experience letter" being a legal requirement associated with your work visa, you could try to explain the reasons why you could not give 4 weeks notice as Judy mentions in her answer. It is possible you could work something out with the employer such as working some extra hours, agreeing to train the new employee, etc.. You would need a good reason for leaving for this to work. If that does not work out, then your next step is to request a copy of your personnel file. The personnel file will probably provide some documentation indicating that you worked for that company and what your duties were. Illinois law requires the employer to provide you with a copy of your personnel file when you make a written request for this to the HR dept.